Learn how to simplify these seemingly devilishly complicated Quant problems!

First, consider these problems

**Q1**.

A.

B.

C.

D.

E.

A.

B.

C.

D.

E.

I.

II.

III.

Rank these three quantities from least to greatest.

A. I,II,III

B. I,III,II

C. II,I,III

D. II,III,I

E. III,I,II

A.

B.

C.

D.

E.

These are challenging problems, especially the third one! With a few simple insights about factorials, though, you will be able to manage all of these.

First of all, a few basics. The factorial is a function we can perform on any positive integer. The expression

It's a good idea to have the first five memorized, simply because those come up frequently on the QUANT — the first five are pretty easy to figure out on your own anyway. ** Nobody** expects you to have the last five here memorized. I give them here purely to give you a sense of how quickly the factorials grow. By the time we get to

**Big Idea #1: every factorial is a factor of every higher factorial**

The number

**Big Idea #2: you can "unpack" one factorial down to another.**

Think about

Well, by the Associative Law, we can group factors in any groupings, so we could insert parenthesis wherever we like. In particular, if I include a selection of factors that goes all the way to the right, all the way to 1, that's another factorial. Thus:

I chose

That's an "**unpacking**" of the first two factors of **unpacking**", is my own creation: you will not see this term used anywhere else.)

**Big Idea #3: when you divide two factorials, you "unpack" the larger one, and cancel it with the smaller one.**

Example:

**Big Idea #4: when you add/subtract two or more factorials, you unpack them all down to the lowest one, and factor out that common factor.**

For more on "**factoring out**", see the section "**Distributing and factor out**" in this post.

Examples:

Having read these rules, give those three practice problems another try before reading the solutions below.

**Q1**. We are going to use the "factoring out" trick in both the numerator and the denominator:

Now, "**unpack**" that top factorial, to cancel the smaller one in the denominator:

**Answer = D**

**Q2**. "**Unpack**" the factorials in the numerator, so that everything is expressed as a product involving

**Answer = D**

**Q3**. Let's consider the three expressions separately.

For expression I, we merely have to "**Unpack**" the

So, expression I has a value of 49.

The expression II is tricky:

That's a whole lot of factors in the numerator! That numerator is fantastically big: it has forty factors from

The expression III is a little easier than II:

That's also a very big number. Each of the seven factors in the numerator is greater than

Thus, from least to greatest, the order is I, III, II.

**Answer = B**

Just as a note, if you are familiar with the idea of combinations, you may recognize expression II as a combinations number:

That would be the number of unique sets of 7 we could select from a pool of

That's a reasonably big number – just over

This is a larger number than all humans and all other living things (animals & plants & all the way down to single-cell critters), including those alive now as well as those who have ever been alive on Earth. This number is larger than all the money in the world in pennies. This is more than the number of individual atoms comprising planet Earth and everything on Earth. This number is much much larger than the number of stars & planets & pulsars & quasars & black holes & whatever other star-like things in all galaxies & clusters in the visible Universe. The phrase "inconceivably big" does not even begin to capture how big this number is!

**Q4**.